Algebraic calculation involving monomials

One monomial is an algebraic term formed by a number, a variable, or by a multiplication between numbers and variables.

The numerical part of the monomial is called the coefficient and the part composed of variables is called the literal part. For example, in the monomial 2xy the coefficient is 2 and the literal part is xy.

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See below how to algebraic calculation involving monomials.

Addition and subtraction of monomials

A addition or subtraction of monomials is made only between monomials that have the same literal part. When they are, we add or subtract the coefficients and keep the literal part.

Example:

Perform addition and subtraction operations between monomials.

The) $\dpi{120} \mathrm{2x^2 + 5x^2 - 3x^2 }$

The literal part of all three monomials is $\dpi{120} \mathrm{x^2}$ , then we perform the operations between the coefficients and keep the literal part:

$\dpi{120} \mathrm{2x^2 + 5x^2 - 3x^2 }$

$\dpi{120} \mathrm{ (2 + 5 - 3)x^2}$

$\dpi{120} \mathrm{ 4x^2}$

B) $\dpi{120} \mathrm{10ab - 8ab^2 + ab - 6ab^2 + 2a}$

Not all terms have the same literal part, so we perform operations only between the coefficients of those that do:

$\dpi{120} \mathrm{10ab - 8ab^2 + ab - 6ab^2 + 2a }$

$\dpi{120} \mathrm{ (10 + 1)ab +(-8 -6)ab^2 + 2a }$

$\dpi{120} \mathrm{ 11ab-14ab^2 + 2a}$

Multiplication of monomials

Amultiplication of monomials is done by multiplying the coefficients and multiplying the literal parts, whether they are equal or not.

However, if the literal parts are powers with the same base, we use the following property of potentiation: $\dpi{120} \mathrm{x^a\cdot x^b x^{a+b}}$ .

Example:

Multiply between monomials.

The) $\dpi{120} \mathrm{3x\cdot 2y\cdot 6z}$

We multiply the coefficients: $\dpi{120} 3\cdot 2\cdot 6 36$

We multiply the literal parts: $\dpi{120} \mathrm{x\cdot y\cdot z xyz}$

Therefore:

$\dpi{120} \mathrm{3x\cdot 2y\cdot 6z 36xyz}$

B) $\dpi{120} \mathrm{5x^2y\cdot 2ax^3y}$

We multiply the coefficients: $\dpi{120} 5\cdot 2 10$

We multiply the literal parts: $\dpi{120} \mathrm{x^2y\cdot ax^3y ax^{2+3}y^{1+1} ax^5y^2}$

Therefore:

$\dpi{120} \mathrm{5x^2y\cdot 2ax^3y 10ax^5y^2}$

division of monomials

At division of monomials, we must divide between the coefficients and between the literal parts of the same base, using another power property: $\dpi{120} \mathrm{x^a: x^b x^{a-b}}$ .